A138591 -id:A138591 - cp's OEIS frontend

A174070 Numbers that can be written as a sum of at least 3 consecutive squares.

14, 29, 30, 50, 54, 55, 77, 86, 90, 91, 110, 126, 135, 139, 140, 149, 174, 190, 194, 199, 203, 204, 230, 245, 255, 271, 280, 284, 285, 294, 302, 330, 355, 365, 366, 371, 380, 384, 385, 415, 434, 446, 451, 476, 492, 501, 505, 506, 509, 510, 534, 559, 590, 595

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

nonn

Numbers of the form (a(a+1)(2a+1)-b(b+1)(2b+1))/6 where a >= b+3 and b >= 0. - Robert Israel, Jul 18 2017

			14 = 1^2 + 2^2 + 3^2, 29 = 2^2 + 3^2 + 4^2.
30 = 1^2 + 2^2 + 3^2 + 4^2, 50 = 3^2 + 4^2 + 5^2.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A111774, A138591, A174069.

N:= 1000: # to get all terms <= N
R:= [seq(b*(b+1)*(2*b+1)/6, b=0..ceil(sqrt(N/3)))]:
sort(convert(select(`<=`, {seq(seq(R[i]-R[j],j=1..i-3),i=1..nops(R))},N),list)); # Robert Israel, Jul 18 2017

max=50^2;lst={};Do[z=n^2+(n+1)^2;Do[z+=(n+x)^2;If[z>max,Break[]];AppendTo[lst,z],{x,2,max/2}],{n,max/2}];Union[lst]
(* Second program: *)
Function[s, Function[t, Union@ Flatten@ Map[TakeWhile[#, # < t[[1, -1]] &] &, t]]@ Map[Total /@ Partition[s, #, 1] &, Range[3, Length@ s]]][Range[16]^2] (* Michael De Vlieger, Jul 18 2017 *)
Module[{nn=30,sq},sq=Range[nn]^2;Take[Union[Flatten[Table[Total/@ Partition[ sq,n,1],{n,3,nn-2}]]],2nn]] (* Harvey P. Dale, Nov 16 2017 *)

A174071 Numbers that can be written as a sum of at least 4 consecutive positive squares.

Original entry on oeis.org

30, 54, 55, 86, 90, 91, 126, 135, 139, 140, 174, 190, 199, 203, 204, 230, 255, 271, 280, 284, 285, 294, 330, 355, 366, 371, 380, 384, 385, 415, 446, 451, 476, 492, 501, 505, 506, 510, 534, 559, 595, 615, 620, 630, 636, 645, 649, 650, 679, 728, 730, 734, 764

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 06 2010

nonn

Numbers of the form m*(6*k^2 + 6*k*m + 2*m^2 - 6*k - 3*m + 1)/6 for some m>=4 and k>=1. - Robert Israel, May 06 2019

			30=1^2+2^2+3^2+4^2, 54=2^2+3^2+4^2+5^2, 55=1^2+2^2+3^2+4^2+5^2, ...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A111774, A138591, A174069, A174070, A307937.

N:= 1000: # to get all terms <= N
Res:= NULL:
for m from 4 while m*(m+1)*(2*m+1)/6 <= N do
   for k from 1 do
       v:= m*(6*k^2 + 6*k*m + 2*m^2 - 6*k - 3*m + 1)/6;
       if v > N then break fi;
       Res:= Res, v;
od od:
sort(convert({Res},list));  Robert Israel, May 06 2019

max=60^2;lst={};Do[z=n^2+(n+1)^2+(n+2)^2;Do[z+=(n+x)^2;If[z>max,Break[]];AppendTo[lst,z],{x,3,Sqrt[max]/2}],{n,Sqrt[max]/2}];Union[lst]

Edited by Robert Israel, May 06 2019

A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

Original entry on oeis.org

5, 6, 9, 12, 13, 14, 15, 17, 18, 21, 22, 24, 25, 27, 28, 29, 30, 33, 35, 36, 37, 38, 39, 41, 42, 44, 45, 46, 48, 49, 51, 53, 54, 55, 56, 57, 60, 61, 62, 63, 65, 66, 69, 70, 72, 73, 75, 76, 77, 78, 81, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 101, 102

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 12 2010

nonn

Numbers of the form sum_{i=j..2k+1} i where j>=1 and 2k+1>j and k>=1. Numbers of the form (2k+1+j)*(2k+2-j)/2, j>=1, k>=1, 2k+1>j. - R. J. Mathar, Dec 04 2011
Subsequences include the A000384 where >=6, the A014106 where >=5, A071355 where >=12, A130861 where >=9, A139577 where >=13, A139579 where >=17 etc. The sequence is the union of all odd-indexed rows of A141419, except its first column and numbers <=3: {5,6}, {9,12,14,15}, {13,18,22,25,27,28}, ... - R. J. Mathar, Dec 04 2011
Does this sequence have asymptotic density 1? - Robert Israel, Nov 27 2018

			5=2+3, 6=1+2+3, 9=4+5, 12=3+4+5,...

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A177712, A136724, A177713.
Cf. A000384, A014106, A071355, A130861, A139577, A139579, A141419.
Contains A004766, A017137 and nonzero terms of A008588.
Disjoint from A002145.
Subsequence of A138591.

f:= proc(n) local r,k;
  for r in select(t -> (2*t-1)^2 >= 1+8*n, numtheory:-divisors(2*n) minus {2*n}) do
    k:= (r + 2*n/r - 3)/4;
    if k::posint and r >= 2*k+2 then return true fi
  od:
  false
end proc:
select(f, [$1..1000]); # Robert Israel, Nov 27 2018

z=200;lst1={};Do[c=a;Do[c+=b;If[c<=2*z,AppendTo[lst1,c]],{b,a-1,1,-1}],{a,1,z,2}];Union@lst1

A163172 Maxima in A163169.

Original entry on oeis.org

0, 1, 6, 10, 20, 28, 44, 88, 104, 136, 272, 304, 368, 464, 496, 592, 1184, 1312, 1376, 1504, 1696, 1888, 1952, 2144, 4288, 4544, 4672, 5056, 5312, 5696, 6208, 6464, 6592, 6848, 6976, 7232, 8128, 8384, 16768, 17536, 17792, 19072, 19328, 20096, 20864

Offset: 1

Carl R. White, Jul 22 2009, Jul 23 2009

nonn

Ray Chandler, Table of n, a(n) for n = 1..10000

Cf. A163169, A138591, A057716.

A165514 The complement of the trapezoidal numbers.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 10, 16, 28, 32, 64, 128, 136, 256, 496, 512, 1024, 2048, 4096, 8128, 8192, 16384, 32768, 32896, 65536, 131072, 262144, 524288, 1048576, 2097152, 4194304, 8388608, 16777216, 33550336, 33554432, 67108864, 134217728, 268435456, 536870912, 1073741824

Offset: 1

Ant King, Sep 23 2009

nonn

Trapezoidal numbers (A165513) are polite numbers (A138591) that have a runsum representation which excludes one, and hence that can be depicted graphically by a trapezoid. This sequence is their complement, and Jones and Lord have shown that it is constructed from the powers of 2 (A000079), the perfect numbers (A000396) and those integers of the form 2^(k-1)*(2^k+1) where k is necessarily a power of 2 and 2^k + 1 is a Fermat prime (A019434).
Starting with 4, composite numbers (A002808) not a difference of non-neighboring triangular numbers (A000217). For T(x) - T(y), x - y > 1, where T are the triangular numbers, all other composite numbers can be represented as a triangular number difference. - Ed Pegg Jr, Feb 23 2016
It appears that these are also the numbers k with the property that all noncentral widths of the symmetric representation of sigma(k) are 1's, with a(1) = 1. Omar E. Pol, Mar 04 2023

			As the fifth integer which does not have a runsum representation which excludes one is 6, then a(5)=6.

Chai Wah Wu, Table of n, a(n) for n = 1..2577
Chris Jones and Nick Lord, Characterizing Non-Trapezoidal Numbers, The Mathematical Gazette, Vol. 83, No. 497, July 1999, pp. 262-263.
Ron Knott, Introducing Runsums
Jim Smith, Trapezoidal numbers, Mathematics in School, Vol. 26, No. 5 (Nov., 1997), pp. 46-47.
T. Verhoeff, Rectangular and Trapezoidal Arrangements, J. Integer Sequences, Vol. 2, 1999, #99.1.6.

Cf. A138591, A165513, A019434, A000396, A000079, A000217, A002808, A237593.

trapezoidal[n_] := Module[{result}, result = {}; Do[sum = 0; start = i; lis = {}; m = i; While[sum < n, sum = sum + m; lis = AppendTo[lis, m]; If[sum == n, AppendTo[result, lis]]; m++], {i, 2, Floor[n/2]}]; result]; Select[Range[10000], trapezoidal[#] == {} &]

More terms from Amiram Eldar, Aug 12 2019

A342285 Coordination sequence with respect to the central vertex of a dodecagon-based tiling of the plane by copies of a certain Goldberg quadrilateral tile.

Original entry on oeis.org

1, 6, 6, 18, 30, 36, 42, 54, 60, 66, 72, 78, 84, 90, 102, 108, 114, 120, 126, 132, 138, 144, 150, 156, 162, 168, 174, 180, 186, 198, 204, 210, 216, 222, 228, 234, 240, 246, 252, 258, 264, 270, 276, 282, 288, 294, 300, 306, 312, 318, 324, 330, 336, 342, 348, 354, 360

Offset: 0

N. J. A. Sloane, Mar 23 2021

nonn

There are many ways to tile the plane with the Goldberg tile; this is a particularly symmetric one.

In Cye Waldman's drawing, six copies of the gray sector are placed at the degree-4 vertices of the decagon, and 6 copies of a similar sector at the degree-6 vertices of the decagon.

Goldberg, M. (1955). “Central Tessellations,” Scripta Mathematica, 21, pp. 253-260. See Fig.7b.
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987; Fig. 1.3.6(a), page 30.

Rémy Sigrist, Table of n, a(n) for n = 0..2500
Rémy Sigrist, Illustration of initial terms
Rémy Sigrist, PARI program for A342285
N. J. A. Sloane, Illustration of initial terms (For example, the a(3) = 18 vertices at 3 steps from the center are colored blue; the a(4) = 30 vertices at 4 steps from the center are green with a black circle.)
Cye Waldman, The Goldberg Tile
Cye Waldman, Central portion of tiling.
Cye Waldman and others, Numerous postings to the Google Groups Tiling Mailing List about tilings with this Goldberg quadrilateral tile, March 2021.

Cf. A138591, A250120.

PARI
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See Links section.
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Apparently, a(n) = 6*A138591(n-1) for n > 1. - Rémy Sigrist, Mar 30 2021

More terms from Rémy Sigrist, Mar 29 2021

A357618 a(n) = sum of lengths of partitions of more than one consecutive positive integer adding up to n.

Original entry on oeis.org

0, 0, 0, 2, 0, 2, 3, 2, 0, 5, 4, 2, 3, 2, 4, 10, 0, 2, 7, 2, 5, 11, 4, 2, 3, 7, 4, 11, 7, 2, 12, 2, 0, 11, 4, 14, 11, 2, 4, 11, 5, 2, 14, 2, 8, 25, 4, 2, 3, 9, 9, 11, 8, 2, 16, 17, 7, 11, 4, 2, 16, 2, 4, 27, 0, 17, 18, 2, 8, 11, 16

Offset: 0

Daniel Vik, Oct 06 2022

A polite number (A138591) has at least one partition of two or more consecutive positive integers that equals n. This sequence is the sum of lengths of all partitions that make a number polite.

This sequence is similar to A204217 which sums lengths of all partitions adding up to n including the partition of length 1.

			n=15 is the sum of three partitions of n with two or more consecutive positive integers: 15 = 1 + 2 + 3 + 4 + 5, 15 = 4 + 5 + 6, 15 = 7 + 8.
The sum of the lengths of these partitions is a(15) = 5 + 3 + 2 = 10.
On the other hand a(8) = 0 because there are no partitions of two or more consecutive integers adding up to 8.

Wikipedia, Polite number

Cf. A069283 (politeness of a number), A138591 (polite numbers).

Cf. A204217.

def A357618(n):
  i=2;r=0
  while n//i>0:r+=(n%i==1)*i;n-=i;i+=1
  return r
A357618_list = [A357618(n) for n in range(70)]

a(n) = A204217(n) - 1 for n >= 1, a(0) = 0.

A138593 Sums of 5 or more consecutive nonnegative integers.

Original entry on oeis.org

10, 15, 20, 21, 25, 27, 28, 30, 33, 35, 36, 39, 40, 42, 44, 45, 49, 50, 51, 52, 54, 55, 56, 57, 60, 63, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 81, 84, 85, 87, 88, 90, 91, 92, 93, 95, 98, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116, 117, 119, 120, 121

Offset: 1

Vladimir Joseph Stephan Orlovsky, May 13 2008

nonn

Also: the numbers of the form s*k+A000217(k-1), s>=0, k>=5. Changing the lower limit to k from 5 to 2, 4 or 6 also defines A138591, A138592 and A138594. [From R. J. Mathar, Jul 31 2008]

			0+1+2=3+4=10, 1+2+3+4+5=15, 2+3+4+5+6=20, 1+2+3+4+5+6=21,...

Ray Chandler, Table of n, a(n) for n = 1..10000

Array[r,9];k=0;For[i=0,i<=55,a=i*4+6;For[j=i+4,j<=66,a=a+j;k++;r[k]=a;j++ ];i++ ];q=Union[Array[r,k]];StringTake[ToString[q],123]

More terms from Vladimir Joseph Stephan Orlovsky, Mar 07 2010

A158754 Courteous prime numbers.

Original entry on oeis.org

31, 41, 101, 109, 137, 191, 193, 239, 283, 311, 313, 569, 641, 659, 661, 821, 1091, 1153, 1289, 1291, 1301, 1303, 1319, 1429, 1453, 1619, 1667, 1877, 2027, 2081, 2129, 2239, 2309, 2341, 2381, 2591, 2593, 2711, 2713, 2971, 3001, 3121, 3253, 3299, 3301

Offset: 1

Vladimir Joseph Stephan Orlovsky, Mar 25 2009

nonn

Courteous prime numbers are twin prime numbers and can be written as the sum of two or more consecutive twin prime numbers. 7+11+13=31,11+13+17=41,..

Cf. A138591, A065867

lst={};Do[pp=Prime[m];If[PrimeQ[pp-2]||PrimeQ[pp+2],k=1;Do[k++;p=Prime[n];If[PrimeQ[p-2]||PrimeQ[p+2],pp+=p;If[PrimeQ[pp],If[PrimeQ[pp-2]||PrimeQ[pp+2],AppendTo[lst,pp]]]],{n,m+1,4*6!}]],{m,1,4*6!}];lst;(*Courteous Prime Numbers.*)Take[Union[lst],5! ]
upto=3400;With[{tps=Union[Flatten[Select[Partition[Prime[Range[ PrimePi[ upto]]], 2,1], Last[#]-First[#]==2&]]]}, Select[Union[Flatten[Table[ Total/@Partition[tps,n,1],{n,2,40}]]],MemberQ[tps,#]&&#<=upto&]] (* Harvey P. Dale, Sep 25 2011 *)

A256504 Summative Fission - For a positive integer n, find the greatest number of consecutive positive integers (at least 2) which add to n. For each of these do the same ... iterate to completion. a(n) = the total number of integers (including n itself) defined.

Original entry on oeis.org

0, 1, 1, 3, 1, 5, 6, 5, 1, 6, 7, 12, 10, 12, 11, 12, 1, 8, 16, 14, 17, 18, 18, 23, 13, 21, 18, 22, 23, 24, 19, 14, 1, 22, 20, 23, 24, 31, 27, 25, 26, 36, 28, 37, 29, 30, 42, 37, 22, 32, 37, 38, 35, 41, 36, 37, 43, 42, 37, 44, 44, 34, 33, 47, 1, 48, 49, 43, 53

Offset: 0

Gordon Hamilton, Mar 31 2015

The iteration that leads to this sequence is worthy of consideration for the grade 2 classroom learning addition.

a(2^k)=1 for all nonnegative integers k as can be seen from A138591.

			a(23) = 23 because there are 23 numbers generated by the iteration:
                  23
                  /\
                 /  \
                /    \
               /      \
              /        \
             /          \
            /            \
          11             12
          /\             /|\
         /  \           / | \
        /    \         /  |  \
       /      \       3   4   5
      /        \     / \     / \
     5          6   1   2   2   3
    / \        /|\             / \
   2   3      / | \           1   2
      / \    /  |  \
     1   2  1   2   3
                   / \
                  1   2
a(24) = 13 because there are 13 numbers generated by the iteration:
          24
          /|\
         / | \
        /  |  \
       7   8   9
      / \     /|\
     3   4   / | \
    / \     /  |  \
   1   2   2   3   4
              / \
             1   2

Martin Büttner, Table of n, a(n) for n = 0..10000

Cf. A138591.

fission[0] = 0;
fission[n_] := fission@n = Module[{div = SelectFirst[Reverse@Divisors[2 n], (OddQ@# == IntegerQ[n/#] && n/# > (# - 1)/2) &]}, If[div == 1, 1, 1 + Total[fission /@ (Range@div + n/div - (div + 1)/2)]]];
fission /@ Range[0, 100] (* Martin Büttner, Jun 04 2015 *)

Corrected and extended by Martin Büttner, Jun 04 2015

cp's OEIS Frontend

A174070 Numbers that can be written as a sum of at least 3 consecutive squares.

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A174071 Numbers that can be written as a sum of at least 4 consecutive positive squares.

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A177731 Numbers which can be written as a sum of consecutive numbers, where the largest term in the sum is an odd number >= 3.

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A163172 Maxima in A163169.

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A165514 The complement of the trapezoidal numbers.

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Mathematica

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A342285 Coordination sequence with respect to the central vertex of a dodecagon-based tiling of the plane by copies of a certain Goldberg quadrilateral tile.

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A357618 a(n) = sum of lengths of partitions of more than one consecutive positive integer adding up to n.

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A138593 Sums of 5 or more consecutive nonnegative integers.

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